Ch.6:Residue Thoory Ch.6:Residue Theory 6.1 The Residue Theorem Chapter 6:Residue Theory 6.2 Trigonometric Integrals Over (0,2) Li,Yongzhao State Key Laboratory of Integrated Services Networks,Xidian University 6.3 Improper Integrals of Certain Functions Over(-oo,oo) June7,2009 6.4 Improper Integrals Involving Trigonometric Functions Ch.6:Residue Theory Ch.6:Residue Theory 6.1 The Residue Theorem L6.1 The Residue Theorem Introduction The Residue Theorem In the previous chapters,we have seen how the theory of If f(z)is analytic on and inside a simple closed positively contour integration lends great insight into the properties of oriented contour I except a single isolated singularity.zo. analytic functions lying interior to T.f(z)has a Laurent series expansion The goal this chapter is to explore another dividend of this theory,namely,its usefulness in evaluating certain real ()=a(:--o j=-00 integrals converging to some punctured neighborhood of z0 We shall begin by presenting a technique for evaluating contour integrals that is known as residue theory In particular,the above equation is valid for all z on the small positively oriented circle C continuously deformed from I(as Then we will introduce some application of the theory to the shown in Fig.6.1) evaluating the real integralsCh.6: Residue Theory Chapter 6: Residue Theory Li, Yongzhao State Key Laboratory of Integrated Services Networks, Xidian University June 7, 2009 Ch.6: Residue Theory Outline 6.1 The Residue Theorem 6.2 Trigonometric Integrals Over (0, 2π) 6.3 Improper Integrals of Certain Functions Over (−∞, ∞) 6.4 Improper Integrals Involving Trigonometric Functions Ch.6: Residue Theory 6.1 The Residue Theorem Introduction In the previous chapters, we have seen how the theory of contour integration lends great insight into the properties of analytic functions The goal this chapter is to explore another dividend of this theory, namely, its usefulness in evaluating certain real integrals We shall begin by presenting a technique for evaluating contour integrals that is known as residue theory Then we will introduce some application of the theory to the evaluating the real integrals Ch.6: Residue Theory 6.1 The Residue Theorem The Residue Theorem If f(z) is analytic on and inside a simple closed positively oriented contour Γ except a single isolated singularity, z0, lying interior to Γ, f(z) has a Laurent series expansion f(z) = ∞ j=−∞ aj (z − z0)j converging to some punctured neighborhood of z0 In particular, the above equation is valid for all z on the small positively oriented circle C continuously deformed from Γ (as shown in Fig. 6.1)